For any integer $m > 2$, there exists an integer
$k = k(m)$ such that the following holds. If $g_1x+h_1, . . . , g_kx+h_k$ is an admissible
$k$-tuple, then for inﬁnitely many integers $n$, there exist $m$ or more primes among
$g_1n + h_1, . . . , g_kn + h_k$.

Now, Maynard's paper and Tao's blog posts only deal with the case $g_1 = \ldots = g_k = 1$. The author cites the stronger statement from Granville's article http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf, where a result is stated for "every admissible $k$-tuple of linear forms" (Theorem 6.2). It is unclear to me if Granville really means to allow tuples of the form $m_i n + h_i$, or if he means only $n+ h_i$. The proof he gives is certainly in the latter case only, but I was thinking that it might be possible to extend it without much difficulty - I'll come back to this later. My reason for confusion is that on the bottom of page 5, Granville states the theorem for forms like $n+h_i$ only.

I've studied Maynard's paper a bit and it does not seem to me that the arguments go through in a straightforward way at this level of generality. The step that seems to break down to me is in Maynard's Lemma 5.2, where one is estimating

This sum essentially counts the number of primes in the range $[g_mN+h_m, 2g_m N + h_m]$ satisfying a congruence condition modulo $g_m W \prod [d_i, e_i]$. So if we try to apply Bombieri-Vinogradov, we would get roughly the same main term but the error term would be a sum over moduli of the form $g_m q$, like

$$\sum_{q = 1}^{ N} E( g_m N, g_m q). $$

By Bombieri-Vinogradov we can say that this is $\ll \frac{g_m N}{(\log g_m N)^A}$, but if we are allowing the $g_m$ to become arbitrarily large, this error term could swamp the main term. I was thinking that perhaps there is a stronger form of Bombieri-Vinogradov saying that we could divided by $g_m$ in the estimate above, since we are only interested in $\frac{1}{g_m}$ of the moduli, but unfortunately this question seems to already have been asked and answered in the negative at The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$.

All I can salvage for now is a statement like the quoted theorem when one has an a priori bound $|g_i | < B$ (for any $B$), and I am wondering if there is a way to get the full strength of it.

Andrew's survey states explicitly that the arguments extend to arbitrary linear forms (see bottom of page 6). In your notation, the $g_i$ stay fixed as $N \to \infty$, so the error terms from Bombieri-Vinogradov etc. are still acceptable. (Of course, as the $g_i$ get larger, one expects $N$ and hence $n$ to get larger also, but the theorem of James and myself does not specify any bound on these $n$, only that they appear infinitely often.)
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Terry TaoApr 8 '14 at 1:56